On the limit in the CLT for a field of martingale differences with respect to a completely commuting invariant filtration

The now classical convergence in distribution theorem for well normalized sums of stationary martingale increments has been extended to multi-indexed martingale increments. In the present article we make progress in the identification of the limit law. In dimension one, as soon as the stationary martingale increments form an ergodic process, the limit law is normal, and it is still the case for multi-indexed martingale increments when one of the processes defined by one coordinate of the multidimensional time is ergodic. In the general case, the limit may be non normal. In the present paper we establish links between the dynamical properties of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℤ</mml:mi> <mml:mi>d</mml:mi> </mml:msup> </mml:math> -measure preserving action associated to the stationary random field (like the positivity of the entropy of some factors) and the existence of a non normal limit law. The identification of a natural factor on which the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℤ</mml:mi> <mml:mi>d</mml:mi> </mml:msup> </mml:math> -action is of product type is a crucial step in this approach.